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Topology and Verb Classes

Feature Column Archive

5. The cusp: ``The fish swallows the bug." etc.

The cusp catastrophe is exemplified by the minimum of the function   f(x) = x4  at   x = 0 . This critical point is unstable: perturbing the function to   f(x) = x4 + ax2   gives a function with three critical points (two minima) near zero if  a  is negative, no matter how small. The minimum of   f(x) = x4  has an additional instability: the perturbation to   f(x) = x4 + ax2 + bx  will have two minima near zero if   b < (4/3)(-a3/6)1/2  and one otherwise. (For a mechanical instantiation of this catastrophe see The Catastrophe Machine.)

The (a,b) parameter space for the cusp catastrophe. For parameter values   (a,b)  in the region between the red curves, the energy function   f(x) = x4 + ax2 + bx  has two minima. Otherwise it has one.

Folllowing from right to left the one-dimensional section given by the blue line, one of the local minima disappears and is captured by the other. The verb class corresponding to this morphology is capture: The fish swallows the bug.

Following the blue section from left to right, a new local minimum appears beside the old one. The verb class corresponding to this morphology is emission: The bird lays an egg.

Following the green line in either direction, one local minimum is substituted by another. The verb class corresponding to this morphology is transformation: The caterpillar becomes a butterfly.